Hilbert's Syzygy Theorem (see [1]}) asserts that the \(d^{th}\)  syzygy module \(Syz^d(M)\)  of a finitely generated \(\mathbb {C}[z_1, .... z_d]\)  module \(M\)  will always be free.
This means that if we choose a free generating set for \(Syz^d(M)\) , then \(Syz^{d+1}(M)=0\) . As a result the correspoding free resolution will terminate at the \((d+1)^{th}\)  step.
We wish to study \(K:=kernel((z_1,...z_d)^P(\hat{\Delta } - \lambda I_n))\) . If \(L\)  is the image of \((z_1,...z_d)^P(\hat{\Delta } - \lambda I_n)\) , then the rows of this matrix generate \(L\)  and \(K\)  is the corresponding generating set.
By the Hilbert Syzygy Theorem applied to \(L\) , this means that \(Syz^{d-1}(K)\)  is always free and hence choosing a free generating set of \(Syz^{d-1}(K)\) , \(Syz^{d}(K)=0\) .
With all this in mind, we have a formula for the density \(dk(\lbrace \lambda \rbrace )\)  using syzygy modules.
